Slide Copyright © 2009 Pearson Education, Inc. 4.1 Variation

Slide Copyright © 2009 Pearson Education, Inc. Direct Variation Variation is an equation that relates one variable to one or more other variables. In direct variation, the values of the two related variables increase or decrease together. If a variable y varies directly with a variable x, then y = kx where k is the constant of proportionality (or the variation constant).

Slide Copyright © 2009 Pearson Education, Inc. Example The amount of interest earned on an investment, I, varies directly as the interest rate, r. If the interest earned is $50 when the interest rate is 5%, find the amount of interest earned when the interest rate is 7%. I = rx

Slide Copyright © 2009 Pearson Education, Inc. Inverse Variation When two quantities vary inversely, as one quantity increases, the other quantity decreases, and vice versa. If a variable y varies inversely with a variable, x, then where k is the constant of proportionality.

Slide Copyright © 2009 Pearson Education, Inc. Example Suppose y varies inversely as x. If y = 12 when x = 18, find y when x = 21.

Slide Copyright © 2009 Pearson Education, Inc. Joint Variation One quantity may vary directly as the product of two or more other quantities. The general form of a joint variation, where y, varies directly as x and z, is y = kxz where k is the constant of proportionality.

Slide Copyright © 2009 Pearson Education, Inc. Example The area, A, of a triangle varies jointly as its base, b, and height, h. If the area of a triangle is 48 in 2 when its base is 12 in. and its height is 8 in., find the area of a triangle whose base is 15 in. and whose height is 20 in.

Slide Copyright © 2009 Pearson Education, Inc. Combined Variation A varies jointly as B and C and inversely as the square of D. If A = 1 when B = 9, C = 4, and D = 6, find A when B = 8, C = 12, and D = 5. Write the equation.

Slide Copyright © 2009 Pearson Education, Inc. Combined Variation continued Find the constant of proportionality. Now find A.

Slide Copyright © 2009 Pearson Education, Inc. 4.2 Linear Inequalities

Slide Copyright © 2009 Pearson Education, Inc. Symbols of Inequality a < b means that a is less than b. a  b means that a is less than or equal to b. a > b means that a is greater than b. a  b means that a is greater than or equal to b. Change the direction of the inequality symbol when multiplying or dividing both sides of an inequality by a negative number.

Slide Copyright © 2009 Pearson Education, Inc. Example: Graphing Graph the solution set of x  4, where x is a real number, on the number line. The numbers less than or equal to 4 are all the points on the number line to the left of 4 and 4 itself. The closed circle at 4 shows that 4 is included in the solution set.

Slide Copyright © 2009 Pearson Education, Inc. Example: Graphing Graph the solution set of x > 3, where x is a real number, on the number line.

Slide Copyright © 2009 Pearson Education, Inc. Example: Solve and graph the solution Solve 3x – 8 < 10 and graph the solution set.

Slide Copyright © 2009 Pearson Education, Inc. Compound Inequality Graph the solution set of the inequality  4 < x  3 a) where x is an integer. b) where x is a real number

Slide Copyright © 2009 Pearson Education, Inc. Example A student must have an average (the mean) on five tests that is greater than or equal to 85% but less than 92% to receive a final grade of B. Jamal’s grade on the first four tests were 98%, 89%, 88%, and 93%. What range of grades on the fifth test will give him a B in the course?

Slide Copyright © 2009 Pearson Education, Inc. Example continued

Slide Copyright © 2009 Pearson Education, Inc. 4.3 Graphing Linear Equations

Slide Copyright © 2009 Pearson Education, Inc. Rectangular Coordinate System x-axis y-axis origin Quadrant I Quadrant II Quadrant IIIQuadrant IV The horizontal line is called the x-axis. The vertical line is called the y-axis. The point of intersection is the origin.

Slide Copyright © 2009 Pearson Education, Inc. Plotting Points Each point in the xy-plane corresponds to a unique ordered pair (a, b). Plot the point (2, 4). Move 2 units right Move 4 units up 2 units 4 units

Slide Copyright © 2009 Pearson Education, Inc. Graphing Linear Equations Graph the equation y = 5x + 2

Slide Copyright © 2009 Pearson Education, Inc. Graphing Using Intercepts The x-intercept is found by letting y = 0 and solving for x. Example: y =  3x + 6 The y-intercept is found by letting x = 0 and solving for y. Example: y =  3x + 6

Slide Copyright © 2009 Pearson Education, Inc. Example: Graph 3x + 2y = 6

Slide Copyright © 2009 Pearson Education, Inc. Slope The ratio of the vertical change to the horizontal change for any two points on the line.

Slide Copyright © 2009 Pearson Education, Inc. Types of Slope Positive slope rises from left to right. Negative slope falls from left to right. The slope of a vertical line is undefined. The slope of a horizontal line is zero. zero negative undefined positive

Slide Copyright © 2009 Pearson Education, Inc. Example: Finding Slope Find the slope of the line through the points (5,  3) and (  2,  3).

Slide Copyright © 2009 Pearson Education, Inc. The Slope-Intercept Form of a Line Slope-Intercept Form of the Equation of the Line y = mx + b where m is the slope of the line and (0, b) is the y-intercept of the line.

Slide Copyright © 2009 Pearson Education, Inc. Graphing Equations by Using the Slope and y-Intercept Solve the equation for y to place the equation in slope-intercept form. Determine the slope and y-intercept from the equation. Plot the y-intercept. Obtain a second point using the slope. Draw a straight line through the points.

Slide Copyright © 2009 Pearson Education, Inc. Example Graph 2x  3y = 9. Write in slope-intercept form.

Slide Copyright © 2009 Pearson Education, Inc. Horizontal Lines Graph y =  3. Vertical Lines Graph x =  3.